Question

Give two irrational numbers so that their.
(i) sum is an irrational number.
(ii) sum is not an irrational number.
(iii) difference is an irrational number.
(iv) difference is not an irrational number.
(v) product is an irrational number.
(vi) product is not an irrational number.
(vii) quotient is an irrational number.
(viii) quotient is not an irrational number.

Answer 1

(i) Consider the two irrational number $2+\sqrt{3}$ and $\sqrt{3}-2$.
Their sum $=2+\sqrt{3}+\sqrt{3}-2=2\sqrt{3}$ is an irrational number.
(ii) Consider the two irrational numbers $\sqrt{2}$ and $-\sqrt{2}$.
Their sum $=\sqrt{2}+(-\sqrt{2})=0$ is a rational number.
(iii) Consider the two irrational numbers $\sqrt{3}$ and $\sqrt{2}$.
Their difference$=\sqrt{3}-\sqrt{2}$ is an irrational number.
(iv) Consider the two irrational numbers $5+\sqrt{3}$ and $\sqrt{3}-5$.
Their difference $=(5+\sqrt{3})-(\sqrt{3}-5)=10$ is a rational number.
(v) Consider the two irrational numbers $\sqrt{3}$ and $\sqrt{5}$.
Their product $=\sqrt{3}\times \sqrt{5}=\sqrt{15}$ is an irrational number.
(vi) Consider the two irrational numbers $\sqrt{18}$ and $\sqrt{2}$.
Their product $=\sqrt{18}\times \sqrt{2}=\sqrt{36}=6$ is a rational number.
(vii) Consider the two irrational numbers $\sqrt{15}$ and $\sqrt{3}$.
Their quotient $=\frac{\sqrt{15}}{\sqrt{3}}=\sqrt{\frac{15}{3}}=\sqrt{5}$ is an irrational number.
(viii) Consider the two irrational numbers $\sqrt{75}$ and $\sqrt{3}$.
Their quotient $=\frac{\sqrt{75}}{\sqrt{3}}=\sqrt{\frac{75}{3}}=5$ is a rational number.